Question:
Let $E\subset \mathbb{R^n}$ be an open set. If $f:E\to \mathbb{R}$ is $\mathcal{C}^2$, and has the following property: $$D_{11}f+D_{22}f+\dots + D_{nn}f < 0 \quad \forall x \in E$$ show that it can't have any local minima in $E$. You can use the first and second derivative tests for single variable functions.
Solution:
I'm unsure how to make what I have rigorous, but my idea would be to say that if we take the restriction of $f$ to each "axis", we would have single-valued functions. Were we to have a minimum at a point x, then these functions would have themselves a minimum at this point and $D_{ii}f|_{x_i}>0$. Then, the sum of these would be positive, not negative as stated in the question. Is this rigorous?